Loop Scaling Laws Research Articles

A study of an X-ray flare on the solar analogue V895 Tau

Using observations with XMM-Newton, we study the characteristics of a flare emanating from a solar analogous V895 Tau. At the peak of the flare, its luminosity reached 3.3×1030ergs−1, which is ∼ 600 times more energetic than the X10 class flare on the Sun. The quiescent state corona of V895 Tau is depicted by a two-temperature plasma model with temperatures of 3.9 and 11 MK. The flare’s evolution was carefully scrutinized through time-resolved X-ray spectroscopy, unveiling the variations in temperature, emission measure, abundance and luminosity during the flare. The temperature peaked at 36.1 MK, which is approximately four times higher than the pre-flare temperature. Employing a hydrodynamic loop model, we have estimated the half length of the flaring loop to be 5.9×1010 cm. Using the loop scaling laws, other loop parameters like density, pressure, volume, and minimum magnetic field are also estimated, and are found to be similar to those of other flares from similar type of stars.

Generalized Coronal Loop Scaling Laws and Their Implication for Turbulence in Solar Active Region Loops

Recent coronal loop modeling has emphasized the importance of combining both Coulomb collisions and turbulent scattering to characterize field-aligned thermal conduction, which invokes a hybrid loop model. In this work, we generalize the hybrid model by incorporating a nonuniform heating and cross section that are both formulated by a power-law function of temperature. Based on the hybrid model solutions, we construct scaling laws that relate loop-top temperature (T a ) and heating rate (H a ) to other loop parameters. It is found that the loop-top properties for turbulent loops are additionally power-law functions of the turbulent mean free path (λ T ), with the functional forms varying from situation to situation, depending on the specification of the heating and/or areal parameters. More importantly, both a sufficiently footpoint-concentrated heating and a cross-sectional expansion with height can effectively weaken (strengthen) the negative (positive) power-law dependence of T a (H a ) on λ T . The reason lies in a notable reduction of heat flux by footpoint heating and/or cross-sectional expansion in the turbulence-dominated coronal part, where turbulent scattering introduces a much weaker dependence of the conduction coefficient on temperature. In this region, therefore, the reduction of the heat flux predominately relies on a backward flattening of the temperature gradient. Through numerical modeling that incorporates more realistic conditions, this scenario is further consolidated. Our results have important implications for solar active region (AR) loops. With the factors of nonuniform heating and cross section taken into account, AR loops can bear relatively stronger turbulence while still keeping a physically reasonable temperature for nonflaring loops.

Temperature and Differential Emission Measure Profiles in Turbulent Solar Active Region Loops

We examine the temperature structure of static coronal active region loops in regimes where thermal conductive transport is driven by Coulomb collisions, by turbulent scattering, or by a combination of the two. (In the last case collisional scattering dominates the heat transport at lower levels in the loop where temperatures are low and densities are high, while turbulent scattering dominates the heat transport at higher temperatures/lower densities.) Temperature profiles and their corresponding differential emission measure distributions are calculated and compared to observations, and earlier scaling laws relating the loop apex temperature and volumetric heating rate to the loop length and pressure are revisited. Results reveal very substantial changes, compared to the wholly collision-dominated case, to both the loop scaling laws and the temperature/density profiles along the loop. They also show that the well-known excess of differential emission measure at relatively low temperatures in the loop may be a consequence of the flatter temperature gradients (and so increased amount of material within a specified temperature range) that results from the predominance of turbulent scattering in the upper regions of the loop.

Coronal Loop Scaling Laws for Various Forms of Parallel Heat Conduction

Abstract The solar atmosphere is dominated by loops of magnetic fluxes that connect the multi-million degree corona to the much cooler chromosphere. The temperature and density structure of quasi-static loops are determined by the continuous flow of energy from the hot corona to the lower solar atmosphere. Loop scaling laws provide relationships between global properties of the loop (such as the peak temperature, pressure, and length); they follow from the physical variable dependencies of various terms in the energy equation, and, hence, the form of the loop scaling law provides insight into the key physics that control the loop structure. Traditionally, scaling laws have been derived under the assumption of collision-dominated thermal conduction. Here, we examine the impact of different regimes of thermal conduction—collision-dominated, turbulence-dominated, and free-streaming—on the form of the scaling laws relating the loop temperature and heating rate to its pressure and half-length. We show that the scaling laws for turbulence-dominated conduction are fundamentally different than those for collision-dominated and free-streaming conduction, inasmuch as the form of the scaling laws now depend primarily on conditions at the low-temperature, rather than high-temperature, part of the loop. We also establish regimes in the temperature and density space in which each of the applicable scaling laws prevail.

AN X-RAY FLARE FROM 47 CAS

Using XMM-Newton observations, we investigate properties of a flare from the very active but poorly known stellar system 47 Cas. The luminosity at the peak of the flare is found to be 3.54 × 1030 , which is ∼2 times higher than that at a quiescent state. The quiescent state corona of 47 Cas can be represented by two temperature plasma: 3.7 and 11.0 MK. The time-resolved X-ray spectroscopy of the flare show the variable nature of the temperature, the emission measure, and the abundance. The maximum temperature during the flare is derived as 72.8 MK. We infer the length of a flaring loop to be 3.3 × 1010 cm using a hydrodynamic loop model. Using the RGS spectra, the density during the flare is estimated as 4.0 × 1010 cm−3. The loop scaling laws are also applied when deriving physical parameters of the flaring plasma.

THE HYDRODYNAMIC EVOLUTION OF IMPULSIVELY HEATED CORONAL LOOPS: EXPLICIT ANALYTICAL APPROXIMATIONS

We derive simple analytical approximations (in explicit form) for the hydrodynamic evolution of the electron temperature T(s, t) and electron density n(s, t), for one-dimensional coronal loops that are subject to impulsive heating with subsequent cooling. Our analytical approximations are derived from first principles, using (1) the hydrodynamic energy balance equation, (2) the loop scaling laws of Rosner–Tucker–Vaiana and Serio, (3) the Neupert effect, and (4) the Jakimiec relationship. We compare our analytical approximations with 56 numerical cases of time-dependent hydrodynamic simulations from a parametric study of Tsiklauri et al., covering a large parameter space of heating rates, heating timescales, heating scale heights, loop lengths, for both footpoint and apex heating, mostly applicable to flare conditions. The average deviations from the average temperature and density values are typically ≈20% for our analytical expressions. The analytical approximations in explicit form provide an efficient tool to mimic time-dependent hydrodynamic simulations, to model observed soft X-rays and extremeultraviolet light curves of heated and cooling loops in the solar corona and in flares by forward fitting, to model microflares, to infer the coronal heating function from light curves of multi-wavelength observations, and to provide physical models of differential emission measure distributions for solar and stellar flares, coronae, and irradiance.

First 3D Reconstructions of Coronal Loops with theSTEREO A+BSpacecraft. II. Electron Density and Temperature Measurements

Using the stereoscopically derived three-dimensional (3D) geometry of 30 loops observed with STEREO EUVI (described in Paper I) we determine here the electron density profiles ne(s) and electron temperature profiles Te(s) from a triple-filter analysis of the stereoscopic images taken in the wavelengths of λ = 171, 195, and 284 A. The statistical results of our analysis of seven complete loops are: observed loop widths wobs = 2.6 ± 0.1 Mm, corresponding to effective loop widths of w = 1.1 ± 0.3 Mm if corrected for the instrumental point-spread function; loop flux ratios floop/ftotal = 0.11 ± 0.04; mean loop (DEM peak) temperatures Tp = 1.1 ± 0.2 MK; DEM temperature Gaussian widths σDEM = 0.35 ± 0.04 MK; temperature variations along loops σT/Tp = 0.24 ± 0.05; (resolution-corrected) loop base densities ne = (2.2 ± 0.5) × 109 cm−3; loop lengths of L = 130 ± 67 Mm; and all quantities are found to agree between STEREO A and B within a few percent. The temperature profiles T(s) along loops are found to be nearly constant, within the uncertainties of the background subtraction. The density profiles ne(s) are consistent with the gravitational stratification of hydrostatic loops, ne(h) = nbaseexp (− h/λT) , defined by the temperature scale heights λT and stereoscopically measured from the height profiles h(s) . The stereoscopic 3D reconstruction allows us for the first time to accurately measure the loop length L and to test loop scaling laws. We find that the observations are not consistent with equilibrium solutions, but rather display the typical overpressures of loops that have been previously heated to higher temperatures and cool down in a nonequilibrium state, similar to earlier EIT and TRACE measurements.

First Three‐Dimensional Reconstructions of Coronal Loops with theSTEREO AandBSpacecraft. I. Geometry

We present one of the first triangulations and 3D reconstructions of coronal loops, using the EUVI telescopes of the two STEREO A and B spacecraft. The first triangulation of coronal loops was performed in an active region, observed with STEREO A and B on 2007 May 9 with a spacecraft separation angle of αsep = 7.3°, at a wavelength of 171 Å. We identify 30 loop structures (7 complete loops and 23 partial segments) and compute their 3D coordinates (x,y,z) (the full 3D coordinates are available as an electronic file). We quantify the height range, the stereoscopic height measurement errors, the loop plane inclination angles, and the coplanarity and circularity of the analyzed loops. The knowledge of the exact 3D geometry of a loop with respect to the observer's line of sight has important consequences for determining the correct vertical density scale height (used in hydrostatic models), the aspect angle of loop cross sections (used in inferring electron densities from optically thin emission measures), the absolute flow speeds (used in siphon flow models), the correct loop length (used in loop scaling laws), and the 3D vectors of the coronal magnetic field (used in testing theoretical magnetic field extrapolation models). The hydrodynamic and magnetic modeling of the analyzed loops will be described in subsequent papers.

X-ray flares on the UV Ceti-type star CC Eridani: a “peculiar” time-evolution of spectral parameters

Context. Weak flares are supposed to be an important heating agent of th e outer layers of stellar atmospheres. However, due to instrumental limitations, only large X-ray flares have been studied in det ail until now. Aims. We used an XMM-Newton observation of the very active BY-Dra type binary star CC Eri in order to investigate the properties of two flares that are weaker than those typically studied in the lit erature. Methods. We performed time-resolved spectroscopy of the data taken with the EPIC-PN CCD camera. A multi-temperature model was used to fit the spectra. We inferred the size of the flaring loops usi ng the density-temperature diagram. The loop scaling laws were applied for deriving physical parameters of the flaring plasma. We also e stimated the number of loops involved in the observed flares. Results. A large X-ray variability was found. Spectral analysis showed that all the regions in the light curve, including the flare segments, are well-described by a 3-T model with variable emission measures but, surprisingly, with constant temperatures (values of 3, 10 and 22 MK). The analysed flares lasted ∼ 3.4 and 7.1 ks, with flux increases of factors 1.5 ‐ 1.9. They oc curred in arcades made of a few tens of similar coronal loops. The size of the flaring loops is much smaller than the distance between the stellar surfaces in th e binary system, and even smaller than the radius of each of the stars. The obtained results are consistent with the following ideas: (i) th e whole X-ray light curve of CC Eri could be the result of a superposition of multiple low-energy flares, and (ii) stellar flares can be scaled-u p versions of solar flares.

Calculating the Thermal Structure of Solar Active Regions in Three Dimensions

We describe a technique to obtain the temperature and density distribution in an active region for a specified plasma heating model. The technique can be applied in general to determine the magnetic field and thermal structure self-consistently. For simplicity, we illustrate the application of this technique in the limit of small plasma β, in which the plasma dynamics decouples from that of the magnetic field, a good approximation in active regions, in which the magnetic field is strong. We select a particular active region, observed in 1996 August, to demonstrate the methodology. We apply the technique to a force-free magnetic field with a plasma heating model in which the volumetric coronal heating rate is directly proportional to the strength of the local magnetic field, and we compute the expected extreme-ultraviolet and soft X-ray emissions from the resulting thermal structure. We compare our solutions with one-dimensional loop models and analytic loop scaling laws. In the future, we plan to compare these emission images with those obtained by the SOHO EUV Imaging Telescope (EIT) and the Yohkoh Soft X-Ray Telescope (SXT) and to explore the relationship between coronal emission and various coronal heating models.

CDS/SoHO multi-line observation of a solar active region: Detection of a hot stable loop and of a cool dynamic loop

We analyze a space-, time- and spectral-resolved SoHO/CDS observation of the evolution of an active region over a time lapse of approximately three hours in various spectral lines emitted in the interval of temperature 1.3 x 10 4 < T < 2.5 x 10 6 K. We identify and characterize two structures of interest: a longer coronal loop (5.5 × 10 9 cm), relatively steady and well visible in lines forming at coronal temperatures (e.g. Fe XIV 334.17 A, Fe XVI 360.76 A) and a smaller one (1.8 × 10 9 cm), transient and visible only in cooler lines (O IV 554.51 A, O V 629.73 A). In the hot lines, the longer loop has a bright apex and an emission distribution of constant shape, but of moderately variable absolute intensity; the region around the loop apex shows a distinct brightening practically in all lines. In the hot lines, the brightening appears as a minor perturbation over a steadily high emission level. In the same region the emission measure vs temperature of the hottest lines indicates a temperature of ∼2 MK, lower than the temperature obtained from Yohkoh data taken just before the CDS observation. Comparison with steady-state loop scaling laws and with plasma time scales, and connection to cooling or heating episodes are discussed. As for the cool loop, its whole evolution, from ignition to disappearance, is directly observed, confirming the highly transient nature of such structures. The O V line is blue-shifted at one footpoint, indicating an upflow associated with the loop ignition.

A Transient Heating Model for Coronal Structure and Dynamics

A wealth of observational evidence for flows and intensity variations in nonflaring coronal loops leads to the conclusion that coronal heating is intrinsically unsteady and concentrated near the chromosphere. We have investigated the hydrodynamic behavior of coronal loops undergoing transient heating with one-dimensional numerical simulations in which the timescale assumed for the heating variations (3000 s) is comparable to the coronal radiative cooling time and the assumed heating location and scale height (10 Mm) are consistent with the values derived from TRACE studies. The model loops represent typical active region loops: 40-80 Mm in length, reaching peak temperatures up to 6 MK. We use ARGOS, our state-of-the-art numerical code with adaptive mesh refinement, in order to resolve adequately the dynamic chromospheric-coronal transition region sections of the loop. The major new results from our work are the following: (1) During much of the cooling phase, the loops exhibit densities significantly larger than those predicted by the well-known loop scaling laws, thus potentially explaining recent TRACE observations of overdense loops. (2) Throughout the transient heating interval, downflows appear in the lower transition region (T ~ 0.1 MK) whose key signature would be persistent, redshifted UV and EUV line emission, as have long been observed. (3) Strongly unequal heating in the two legs of the loop drives siphon flows from the more strongly heated footpoint to the other end, thus explaining the substantial bulk flows in loops recently observed by the Coronal Diagnostic Spectrometer and the Solar Ultraviolet Measurement of Emission Radiation instrument. We discuss the implications of our studies for the physical origins of coronal heating and related dynamic phenomena.

Prominence Formation by Localized Heating

We describe a model for the formation of the cool condensed material that comprises a coronal filament or prominence. Numerical calculations are presented which demonstrate that large condensations form in a coronal loop if the loop satisfies two key requirements: (1) the loop heating must be localized near the chromospheric footpoints, and (2) the loop must have a dipped geometry in order to support the prominence condensation against gravity. We calculate one-dimensional equilibrium solutions for the equations of force and energy balance assuming optically thin radiative losses and a parameterized form for the coronal heating. This physical situation is modeled as a boundary value problem, which we solve numerically using a B-spline collocation scheme. The relation of our solutions to the well-known loop scaling laws is discussed, and the implications of our model for active region and quiescent prominences are discussed.